Question

Express the trigonometric ratios sin A, sec A and tan A in terms of cot A.


Solution:


cosec2 A – cot2 A = 1
cosec2 A = 1 + cot2 A
cosec2 A = cot2 A + 1​

Answer 1

$\begin{array}{l}\cot a=\frac{\cos A}{\sin A} \\\\\cot ^{2} A=\frac{\cos ^{2} A}{\sin ^{2} A}=\frac{1-\sin ^{2} A}{\sin ^{2} A} \\\\\cot ^{2} A=\frac{1}{\sin ^{2} A}-1 \Rightarrow 1+\cot ^{2} A=\frac{1}{1} \\\\\sin A=\frac{1}{\sqrt{1+\cot ^{2} A}}\end{array}$

$\begin{array}{l}\text { Now,sec } A=\frac{1}{\cos A}=\frac{1}{\sqrt{1-\sin ^{2} A}} \\\\\sec A=\frac{1}{\sqrt{1-\frac{1}{1+\cot ^{2} A}}}=\frac{1}{\sqrt{\frac{\cot ^{2} A}{1+\cot ^{2} A}}} \\\\\sec A=\frac{\sqrt{1+\cot ^{2} A}}{\cot A} \\\\\tan A=\frac{1}{\cot A}\end{array}$